标题： 单值等价性对于Lamé型方程I：有限间隙结构和圆锥球面度量 显示英文标题

标题： Monodromy Equivalence for Lamé-type Equations I: Finite-gap Structures and Cone Spherical Metrics

摘要： 受经典Lamé方程(1.2)的有限间隙结构及其在数学物理中的核心作用的启发，研究了广义Lamé型方程(1.12)。对于基本情况$n=1$，建立了经典Lamé方程(1.18)与广义Lamé型方程(1.19)之间的单值等价性。得到了两个主要应用：(i) 推导出(1.19)的有限间隙结构，并对$\tau\in i\mathbb{R}_{>0}$的谱曲线$\sigma_{1}$和$\sigma_{2}$进行了完整的分类；(ii) 将单值等价性应用于具有三个大锥奇点的圆锥球面度量的构造，每个锥角均超过$2\pi$。显示了一类这样的度量表现出爆破配置，该配置用单值数据明确描述。 显示英文摘要

摘要： Motivated by the finite-gap structure of the classical Lam\'{e} equation (1.2) and its central role in mathematical physics, generalized Lam\'{e}-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy equivalence between the classical Lam\'{e} equation (1.18) and the generalized Lam\'{e}-type equation (1.19) is established. Two main applications are obtained: (i) the finite-gap structure of \ (1.19) is derived, together with a complete classification of the spectral curves $\sigma_{1}$ and $\sigma_{2}$ for $\tau\in i\mathbb{R}_{>0}$; and (ii) the monodromy equivalence is applied to the construction of cone spherical metrics with three large conical singularities, each with cone angle exceeding $2\pi$. A family of such metrics is shown to exhibits a blow-up configuration, which is described explicitly in terms of the monodromy data.